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Analysis of a Composite Laminated Beam

Problem Description

A simple composite laminated beam as shown in Fig.1 is clamped at x=0, and subjected to a 10 kN point force in x-direction and a 10 kN∙mm moment in y-direction at x=1m. Dimensions of the beam and layup of the laminate are given in Fig.1. The laminate is made of 8 layers, each of a continuous fiber reinforced composite. The lamina constants are $$E_1 = 110.5 \text{GPa}, E_2 = E_3 = 13.64 \text{GPa}, G_{12} = G_{13} = 3.92 \text{GPa}, G_{23} = 3.26 \text{GPa}, \nu_{12} = \nu_{13} = 0.329, \nu_{23} = 0.400. $$

Fig1_8layerBeam.png


Software Used

This wiki page is based on Gmsh4SC version 2.0.
A previous resource (Gmsh4SC Tutorial #1) including documents and Gmsh files, and the video are based on Gmsh4SC version 1.3.

Updates in tutorial for Gmsh4SC ver. 2.0:
• Draw point and line elements using commands
• CalculiX boundary conditions


Companion Video

Please refer to the YouTube video by Xin Liu showing the solution procedure using Gmsh4SC version 1.3.


Solution Procedure

Below describes the step-by-step procedure you followed to solve the problem. It is better to accompany this document with the companion youtube video.

1. Input Laminate Geometry and Lamina Material Properties

The structure genome (SG) for this laminated beam is the cross section of the laminate beam, because there is no heterogeneity along beam reference line. Here we input material properties of each layer using orthotropic model, and geometry information directly generated by laminate layup code.

1.1 Create a new .geo file

  • Click FileNew
  • Change the filename in /path/to/current/session/Filename.geo (give a name other than Untitled)

Fig2_newfile.png

1.2 Input lamina properties

  • Click MaterialThermoelasticOrthotropic
  • Input the 9 lamina constants
  • Click AddCloseExit

Fig3_MaterialProp.png

1.3 Input laminate geometry by composite layup

  • Click GeometryCommon SG2D SGBeam sections
  • Click Compositeslayup defineFast generate

Fig4_GenLaminate.png

  • Material: 1 -- MAT1 (for lamina)
  • Layup: [0/90/0/90]s (as shown in Fig.1)
  • ply thickness: 5 (unit: mm, as total thickness for 8 layers is 0.04m)

Fig5_GenLaminate_cont.gifFig5_GenLaminate_cont.gif

  • Click AddClose, and get back to the previous window showing Beam Section – Composites
  • Width: 180 (unit: mm, as shown in Fig.1), and click Add

Fig6_Width.png

The geometry of the 8-layer laminate is generated.
Fig7_8layerGeofinished.png

2. Homogenization: Effective SG Properties Calculation

Now consider the 8 layers as a homogeneous material, we calculate the effective stiffness matrix of the SG by Homogenization command.

2.1 Mesh

  • Click MeshMesh control
  • In the General tab, tick Recombine all triangular meshes, input Element size factor: 0.1, then close this window

Fig8_MeshCtrl.png

  • Click Generate 2D meshGenerate, and you’ll see the mesh generated for 8 layers
  • Click Set order 2

Fig9_8layersMeshed.png
If you want to change mesh properties, e.g., change element size factor, you may reopen Mesh control, input the new number, and then click Generate again to get the new mesh.

2.2 Homogenization

  • Click SwiftCompHomogenizationBeam Model
  • No need to change or input anything in this case
  • Click Save, wait for ~10 seconds, click Run

Fig10_homogenization.png

2.3 Check results and save files

  • Once the homogenization finishes, a plain text file Filename.msh.k containing effective stiffness and compliance matrices of the laminate will pop up.

Fig11_8layerResults.png

  • You may download the files for your own record by the following methods.
  1. Use Import/Export tool on cdmhub. The tool is launched by default for every SwiftComp session.
    Fig12_download.png
  2. Use any FTP tool
    • host: cdmhub.org
    • username: if you forgot your username, it can be found in many ways. For example, if you’re logged on cdmhub webpage, on the upper right corner says “Logged in (your_Username)”. Or, in any launched swiftcomp session, click the bottom grey bar or click ToolsMessage Console, to show message console. On the 9th line it says “Info: Home directory: /home/cdmhub/ your_Username /”.
    • password: same as your log in password to cdmhub webpage.
    • remote site address: /home/cdmhub/your_Username/data/sessions/5-digit_current_session_number
    • current session number can be found above the SwiftComp window, or in the title of the webpage, or in Dashboard – Activity.
  3. Use sftp command line (the following commands work for Mac – Terminal)
    • sftp your_Username@cdmhub.org
    • get /home/cdmhub/your_Username/data/sessions/5-digit_current_session_number/Filename.msh.k local_folder_path
    • feel free to use help or ? to check all available commands
  • I usually download these files after homogenization:
    • `Filename.geo`: geometry without mesh
    • `Filename.msh`: geometry and mesh informations
    • `Filename.msh.k`: results after homogenization

3. Macroscopic Analysis for the Beam

3.1 Create beam geometry

  • Click FileNew, and input a different filename, e.g., /home/.../Beam.geo, same as in Fig.2
  • For Gmsh4SC version 2.0, you need to input the beam geometry by commands
  • Click Input controlEdit file
  • Manually add the following three lines:
    Point(1) = {0,0,0,1.0}: create a point of position {x,y,z} = {0,0,0}, mesh size = 1.0
    Point(2) = {0,0,1000,1.0}: create a point at x=1000 (unit: mm)
    Line(1) = {1,2}: create a line connecting the two points.
  • Click Save, then close text editor

Fig13_BeamGeoCmd.png

  • Click Reload, and you will see the beam geometry on the screen.

Fig14_Beamgeo.png

3.2 Input effective SG properties for the beam

  • Click GeometryAssign structural propertiesBeam, Click Close

Fig15_AssignBeamProp.png

  • Click on the red line, Press e and q

Fig16_AssignBeamProp_2.png

  • In the pop-up browser window, choose Filename.msh.k (the results file in Fig.11 after Homogenization step)

Fig17_AssignBeamProp_3.png
Now the beam is considered as a homogeneous material with effective material properties given in Fig.11.

3.3 Generate mesh for the beam

  • Change the elementary mesh size back to 1.0: Click MeshMesh control, input Element size factor: 1.0
  • Click MeshGenerate 1D meshGenerate

Fig18_BeamMesh.png

3.4 Input file for CalculiX

  • Click CalculiXWrite INP file, to create the input file (Filename.inp) for CalculiX

Fig19_INPfile.png

  • Click Import structural properties, to input mesh information in the INP file

Fig20_importstruct.png

  • Click Define step, to check the input file and add boundary conditions manually

Fig21_openINPfile.png
These are the {x,y,z} values for each mesh node. Notice that Node#1 and Node#2 are the two ends, x=0 and x=1000, while other nodes are increasing by step 1 since the size factor is 1.

  • Input the following 8 lines at the end of the opened file:
    *BOUNDARY
    1,1,6 
    *STEP
    *STATIC
    *CLOAD
    2,1,10000
    2,5,10000
    *END STEP
    

    Fig22_inputBC.png

    • Click Save, close Geany editor, click Run.

    Fig23_RunCCX.png

    • If you’re not familiar with CalculiX, the commands you just input have the following meanings:
      • *BOUNDARY
        1,1,6
        1,_,_ means Node#1 (x=0)
        _,1,6 means fixed degree of freedom 1 through 6.
        Degree of freedom 1,2,3: translation in x,y,z direction;
        Degree of freedom 4,5,6: rotation about x,yz, axis.
      • *CLOAD
        2,1,10000
        2,5,10000
        CLOAD means concentrated load
        2,_,_ means Node#2 (x=1000)
        _,1,10000 means degree of freedom 1 (translation in x direction), 10000N force
        _,5,10000 means degree of freedom 5 (rotation about y axis), 10000N∙mm moment
      • For other types of boundary conditions, you may refer to CalculiX documentation.
      • Detailed introduction for CalculiX commands used in Gmsh4SC package can be found in Chapter 4 of the Gmsh4SC User Manual.
  • 3.5 Check results

    • Click Results, and you may navigate the Filename_sc.dat file to check global responses of the beam, including Displacement ($$u_1, u_2, u_3$$), Rotation ($$\theta_1, \theta_2, \theta_3$$, Strain ($$\gamma_{11}, \kappa_1, \kappa_2, \kappa_3$$), and Stress ($$F_1, M_1, M_2, M_3$$) at nodes (x=0 through 1000) and at Gauss points.

    Fig24_BeamResult.png

    • Now you can find the global responses at each position of the beam (e.g., x = 500 mm) referring to the Node number (e.g., Node#502).

    Fig25_Node502.png

    • At x = 500 mm (Node#502), the beam global responses are found to be:
      $$u_1$$=1.110338E-02, $$u_2$$=6.677899E-18(=0), $$u_3$$=-1.611413E-02
      $$\theta_1$$=-1.770997E-21(=0), $$\theta_2$$=6.445670E-05, $$\theta_3$$=2.671169E-20(=0),
      $$\gamma_{11}$$=2.220675E-05, $$\kappa_1$$=-3.542017E-24(=0), $$\kappa_2$$=1.289142E-07, $$\kappa_3$$=5.342389E-23(=0),
      $$F_1$$=1.000000E+04, $$M_1$$=3.097452E-23(=0), $$M_2$$=1.000000E+04, $$M_3$$=-5.376268E-16(=0)

    Fig26_displacement.pngFig27_rotation.pngFig28_strain.pngFig29_loading.png

    • It’s more convenient to download the Filename_sc.dat file on your local computer and search for these values. See section 2.3 of this wiki page for download instructions.

    4. Dehomogenization: Local Stress Distribution

    With the strain at each node given by CalculiX results, for a specific node (e.g., Node#502 at x=500 mm), we can check the local stresses in the 8 lamina layers.

    4.1 Reopen microstructure geometry file

    • Click FileOpen, and choose the the_8-layer_laminate_file.geo

    Fig30_OpenGeo.png

    • If the 8 layer laminate geometry does not appear on the screen, it is possible that it’s outside the current view. You may bring it back by clicking ToolsManipulatorReset

    Fig31_resetview.png

    4.2 Dehomogenization

    • Click SwiftCompDehomogenizationBeam Model
    • Input displacement (v1, v2, v3) and strain (e11, k1, k2, k3) for Node#502. (values are given in section 3.5)

    Fig32_dehomo.png

    • Click SaveRun, the results are shown on the screen with color bars. You may check the displacement, strain and stress results in the Post-processing panel.

    Fig33_dehomoResults.png

    • Stress $$\sigma_{11}$$ of each layer is given in the following figure.

    Fig34_s11.png

    • Numerical results can be found in filenameVIEW.msh. For example, the $$\sigma_{11}$$ results are given for each node.

    Fig35_s11numbers.png

    4.3 Check results along a line (optional)

    • To get the stress values $$\sigma_{11}$$ along the cross section (e.g., x=0, y varies from -20 to +20), you may use the CutGrid plugin.
      • Click on the triangle beside S11, then click Plugin.
      • Select CutGrid, View[10] (S11), and input X0 = 0, Y0 = -20, Z0 = 0 and X1 = 0, Y1 = 20, Z1 = 0. X2 Y2 Z2 can be any number as we only need two points to define a line.
      • NumPointsU = 100 which is the number of points from {X0, Y0, Z0} to {X1, Y1, Z1}.
      • NumPointsV = 1 which is the number of points from {X0, Y0, Z0} to {X2, Y2, Z2}.
      • Position of these three points are labeled on the geometry

    Fig36_CutGrid.png

    • Click Run, and a new view will appear that only contains S11 along x=0, y= -20 to +20. You may export the new view by clicking the triangle – Export. Numerical results including {x,y,z} of each node and S11 values are recorded in the exported file.

    Fig37_S11acrossY.png


    References

    1. Xin Liu; Wenbin Yu (2017), “Gmsh4SC Tutorial #1: Analysis of a Composite Laminated Beam,” https://cdmhub.org/resources/1280.
    2. Gmsh documentation for geometry commands input
    3. Xin Liu; Wenbin Yu (2017), Gmsh4SC User Manual

    Created on , Last modified on